Question

Six people are sitting around a circular table, and each person has either blue eyes or green eyes. Let x be the number of people sitting next to at least one blue-eyed person, and let y be the number of people sitting next to at least one green-eyed person. How many possible ordered pairs (x,y) are there? (For example, (x,y) = (6,0) if all six people have blue eyes, since all six people are sitting next to a blue-eyed person, and zero people are sitting next to a green-eyed person.)

Answer 1

10

Explanation:

Given that:

Six people are sitting around a circular table

and each person has either blue or green eyes.

So; we are told to represent x as the number of people sitting next to at least one blue-eyed person

and y be the number of people sitting next to at least one green-eyed person.

The objective is to find how many possible ordered pairs (x,y) are there?

The ordered pairs (x,y) i.e ( blue , green) are : ( (6, 0), (6, 2), (6, 4), (5, 3), (5, 5), (3, 3) ).

The above proves the ordered pairs for at least as blue as green.

Now; to determine the ordered pairs for as many green as blue, we will require to go the opposite direction by reversing the values; so we can arrive at a new values of 10 possible outcomes which are as follows:

(6, 0), (6, 2), (6, 4), (5, 3), (5, 5), (3, 3), (3, 5), (4, 6), (2, 6), (0, 6)